It's yellow! - a world covered with sulfur spewed from volcanos,
burning hot inside from intense tidal interactions with Jupiter's
mighty gravitational field... but frigid at the surface.

Last week I talked about something called the "Pythagorean
pentagram". That's a cool name - but it's far from clear who
first discovered this entity, so I started feeling a bit guilty for
using it, and I started wondering what we actually know about
Pythagoras or the mathematical vegetarian cult he supposedly launched.
Tim Silverman pointed me to a scholarly book on the subject:

It turns out we know very little about Pythagoras: a few grains of
solid fact, surrounded by a huge cloud of stories that grows larger
and larger as we move further and further away from the 6th century
BC, when he lived. This is especially true when it comes to his
contributions to mathematics. The infamous pseudohistorian Eric
Temple Bell begins his book "The Magic of Numbers" as follows:

The hero of our story is Pythagoras. Born to immortality
five hundred years before the Christian era began, this
titanic spirit overshadows western civilization. In some
respects he is more vividly alive today than he was in his
mortal prime twenty-five centuries ago, when he deflected
the momentum of prescientific history toward our own
unimagined scientific and technological culture. Mystic,
philosopher, experimental physicist, and mathematician of
the first rank, Pythagoras dominated the thought of his age
and foreshadowed the scientifi mysticisms of our own.

But, there's no solid evidence for any of this, except perhaps
his interest in mysticism and numerology and the incredible
growth of his legend as the centuries pass. We're not even
sure he proved the "Pythagorean theorem", much less all the
other feats that have been attributed to him. As Burkert explains:

No other branch of history offers such temptations to
conjectural reconstruction as does the history of mathematics.
In mathematics, every detail has its fixed and unalterable place
in a nexus of relations, so that it is often possible, on the
basis of a brief and casual remark, to reconstruct a complicated
theory. It is not surprising, then, that gap in the history of
mathematics that was opened up by a critical study of the evidence
about Pythagoras has been filled by a whole succession of
conjectural supplements.

There's a new book out on Pythagoras:

3) Kitty Ferguson, The Music of Pythagoras: How an Ancient
Brotherhood Cracked the Code of the Universe and Lit the
Path from Antiquity to Outer Space, Walker and Company, 2008.

The subtitle is sensationalistic, exactly the sort of thing that would
make Burkert cringe. But the book is pretty good, and Ferguson is
honest about this: after asking "What do we know about Pythagoras?",
she lists everything we know in one short paragraph, and then
emphasizes: that's all.

He was born on the island of Samos sometime around 575 BC. He went
to Croton, a city in what is now southern Italy. He died around 495
BC. We know a bit more - but not much.

It's much easier to learn about the Renaissance "neo-Pythagoreans".
This book is a lot of fun, though too romantic to be truly scholarly:

It seems clear that the Renaissance neo-Pythagoreans, and even the
Greek Pythagoreans, and perhaps even old Pythagoras himself were much
taken with something called the tetractys:

o
o o
o o o
o o o o

To appreciate the tetractys, you have to temporarily throw out
modern scientific thinking and get yourself in the mood of
magical thinking - or "correlative cosmology", which tries to
understand the universe by setting up elaborate correspondences
between this, that, and the other thing. To the Pythagoreans,
the four rows of the tetractys represented the point, line, triangle
and tetrahedron. But the "fourness" of the tetractys also
represented the four classical elements: earth, air, water and fire.
It's fun to compare these early groping attempts to impose order
on the universe to later, less intuitive but far more predictively
powerful schemes like the Periodic Table or the Standard Model.
So, let's take a look!

The Renaissance thinkers liked to organize the four elements using
a chain of analogies running from light to heavy:

fire : air :: air : water :: water : earth

Them also organized them in a diamond, like this:

FIRE
hot dry
AIR EARTH
wet cold
WATER

Sometimes they even put a fifth element in the middle: the
"quintessence", or "aether", from which heavenly
bodies were made. And following Plato's Timaeus dialog, they set up
an analogy like this:

This is cute! Fire feels pointy and sharp like tetrahedra, while
water rolls like round icosahedra, and earth packs solidly like
cubes. Dodecahedra are different than all the rest, made of
pentagons, just as you might expect of "quintessence". And
air... well, I've never figured out what air has to do with
octahedra. You win some, you lose some - and in correlative
cosmology, a discrepancy here and there doesn't falsify your ideas.

The tetractys also took the Pythagoreans in other strange directions.
For example, who said this?

"What you suppose is four is really ten..."

A modern-day string theorist talking to Lee Smolin about the dimension
of spacetime? No! Around 150 AD, the rhetorician Lucian of Samosata
attributed this quote to Pythagoras, referring to the tetratkys and the
fact that it has 1 + 2 + 3 + 4 = 10 dots. This somehow led the
Pythagoreans to think the number 10 represented "perfection". If
there turn out to be 4 visible dimensions of spacetime together with
6 curled-up ones explaining the gauge group U(1) × SU(2) × SU(3),
maybe they were right.

Pythagorean music theory is a bit more comprehensible: along with
astronomy, music is one of the first places where mathematical
physics made serious progress. The Greeks, and the Babylonians
before them, knew that nice-sounding intervals in music correspond
to simple rational numbers. For example, they knew that the octave
corresponds to a ratio of 2:1. We'd now call this a ratio of
frequencies; one can get into some interesting scholarly arguments
about when and how well the Greeks knew that sound was a vibration,
but never mind - read Burkert's book if you're interested.

Whatever these ratios meant, the Greeks also knew that a fifth
corresponds to a ratio of 3:2, and a fourth to 4:3.

By the way, if you don't know about musical intervals like
"fourths" and "fifths", don't feel bad. I won't
explain them now, but you can learn about them and hear them here:

If you nose around Capleton's website, you'll see he's quite
a Pythagorean mystic himself!

Anyway, at some moment, lost in history by now, people figured
out that the octave could be divided into a fourth and a fifth:

2/1 = 4/3 × 3/2

And later, I suppose, they defined a whole tone to be the
difference, or really ratio, between a fifth and a fourth:

(3/2)/(4/3) = 9/8

So, when you go up one whole tone in the Pythagorean tuning system,
the higher note should vibrate 9/8 as fast as the lower one. If
you try this on a modern keyboard, it looks like after going up 6
whole tones you've gone up an octave. But in fact if you buy the
Pythagorean definition of whole tone, 6 whole tones equals

(9/8)6 = 531441 / 262144 ≅ 2.027286530...

which is, umm, not quite 2!

Another way to put it is that if you go up 12 fifths, you've
almost gone up 7 octaves, but not quite: the so-called circle
of fifths doesn't quite close, since

(3/2)12 / 27 = 531441 / 524288 ≅ 1.01264326...

This annoying little discrepancy is called the "Pythagorean comma".

This sort of discrepancy is an unavoidable fact of mathematics.
Our ear likes to hear frequency ratios that are nice simple
rational numbers, and we'd also like a scale where the notes are
evenly spaced - but we can't have both. Why? Because you can't
divide an octave into equal parts that are rational ratios of
frequencies. Why? Because a nontrivial nth root of 2 can never
be rational.

So, irrational numbers are lurking in any attempt to create an
equally spaced (or as they say, "equal-tempered") tuning system.

You might imagine this pushed the Pythagoreans to confront
irrational numbers. This case has been made by the classicist
Tannery, but Burkert doesn't believe it: there's no written
evidence suggesting it.

You could say the existence of irrational numbers is
the root of all evil in music. Indeed, the diminished fifth
in an equal tempered scale is called the "diabolus in musica",
or "devil in music", and it has a frequency ratio equal to
the square root of 2.

Or, you could say that this built-in conflict is the spice of
life! It makes it impossible for harmony to be perfect and
therefore dull.

Anyway, Pythagorean tuning is not equal-tempered: it's based
on making lots of fifths equal to exactly 3/2. So, all the
frequency ratios are fractions built from the numbers 2 and 3.
But, some of them are nicer than others:

As you can see, the third, sixth and seventh are not very nice:
they're complicated fractions, so they don't sound great.
They're all a bit sharp compared to the following tuning system,
which is a form of "just intonation":

Just intonation brings in fractions involving the number 5, which
we might call the "quintessence" of music: we need it to get a nice-sounding
third. A long and interesting tale could be told about this tuning system -
but not now. Instead, let's just see how the third, sixth and seventh
differ:

In just intonation the third is 5/4 = 1.25, but in Pythagorean
tuning it's 81/64 = 1.265625. The Pythagorean system is about
1.25% sharp.

In just intonation the sixth is 5/3 = 1.6666.., but in Pythagorean
tuning it's 81/64 = 1.6875. The Pythagorean system is about 0.7%
sharp.

In just intonation the seventh is 15/8 = 1.875, but in Pythagorean
tuning it's 243/128 = 1.8984375. The Pythagorean system is about
1.25% sharp.

Here you can learn more about Pythagorean tuning, and hear it in
action:

There's also a murky relation between Pythagorean tuning and
something called the "Platonic Lambda". This is a certain
way of labelling the edges of the tetractys by powers of 2 on
one side, and powers of 3 on the other:

1
2 3
4 9
8 27

I can't help wanting to flesh it out like this, so going down
and to the left is multiplication by 2, while going down and
to the right is multiplication by 3:

1
2 3
4 6 9
8 12 18 27

So, I was pleased when in Heninger's book I saw the numbers
on the bottom row in a plate from a 1563 edition of "De
Natura Rerum", a commentary on Plato's Timaeus written by
the Venerable Bede sometime around 700 AD!

In this plate, the elements fire, air, water and earth are
labelled by the numbers 8, 12, 18 and 27. This makes the
aforementioned analogies:

fire : air :: air : water :: water : earth

into strict mathematical proportions:

8 : 12 :: 12 : 18 :: 18 : 27

Cute! Of course it doesn't do much to help us understand
fire, air, earth and water. But, it goes to show how people
have been struggling a long time to find mathematical patterns
in nature. Most of these attempts don't work. Occasionally
we get lucky... and over the millennia, these scraps of luck
added up to the impressive theories we have today.

Next: the categorical groups workshop here in Barcelona!

A "categorical group", also called a "2-group", is
a category that's been equipped with structures mimicking those of a
group: a product, identity, and inverses, satisfying the usual laws
either "strictly" as equations or "weakly" as
natural isomorphisms. Pretty much anything people do with groups can
also be done with 2-groups. That's a lot of stuff - so there's a lot
of scope for exploration! There's a powerful group of algebraists in
Spain engaged in this exploration, so it makes sense to have this
workshop here.

Let me say a little about some of the talks we've had so far.
I'll mainly give links, instead of explaining stuff in detail.

Just as we can try to classify principal bundles over some space with
any fixed group as gauge group, we can try to classify "principal
2-bundles" with a given "gauge 2-group". It's a famous
old theorem that for any topological group G, we can find a space BG
such that principal G-bundles over any mildly nice space X are
classified by maps from X to BG. (Homotopic maps correspond to
isomorphic bundles.) A similar result holds for topological 2-groups!

Indeed, Baas Bökstedt and Kro did something much more general
for topological 2-categories:

Just as a group is a category with one object and with all
morphisms being invertible, a 2-group is a 2-category with one
object and all morphisms and 2-morphisms invertible. But
the 2-group case is worthy of some special extra attention,
so Danny Stevenson studied that with a little help from me:

11) John Baez and Danny Stevenson, The classifying space of
a topological 2-group, available as arXiv/0801.3843

and that's what I talked about. If you're also interested in
classifying spaces of 2-categories that aren't topological,
just "discrete", you should try these:

14) Manuel Bullejos, Emilio Faro and Victor Blanco,
A full and faithful nerve for 2-categories, Applied
Categorical Structures 13 (2005), 223-233. Also
available as arXiv:math/0406615.

On Monday afternoon, Bruce Bartlett spoke on a geometric
way to understand representations and "2-representations"
of ordinary finite groups. You can see his talk here, and
also a version which has less material, explained in a
more elementary way:

The first big idea here is that the category of representations
of a finite group G is equivalent to some category where an object
X is a complex manifold on which G acts, equipped with an invariant
hermitian metric and an equivariant U(1) bundle. A morphism from X to
Y in this category is not just the obvious sort of map; instead, it's
diagram of maps shaped like this:

S
/ \
/ \
F/ \G
/ \
v v
X Y

This is called a "span". So, we're seeing a very nice
extension of the Tale of Groupoidification, which began in "week247" and continued up to "week257", when it jumped over to my
seminar.

But Bruce doesn't stop here! He then categorifies this whole
story, replacing representations of G on Hilbert spaces by
representations on 2-Hilbert spaces, and replacing U(1) bundles
by U(1) gerbes. This is quite impressive, with nice applications
to a topological quantum field theory called the Dijkgraaf-Witten
model.

Next, to handle the TQFT called Chern-Simons theory, Bruce plans to
replace the finite group G by a compact Lie group. Another, stranger
direction he could go is to replace G by a finite 2-group. Then he'd
make contact with the categorified Dijkgraaf-Witten TQFT studied in
these papers:

19) João Faria Martins and Timothy Porter, On Yetter's invariant and
an extension of the Dijkgraaf-Witten invariant to categorical groups,
avilable as arXiv:math/0608484.

As the last paper explains, we can also think of this TQFT as a field
theory where the "field" on a spacetime X is a map

f: X → BG

where BG is the classifying space of the 2-group G.

Given all this, it's natural to contemplate a further generalization
of Bruce's work where G is a Lie 2-group. Unfortunately, Lie 2-groups
don't have many representations on 2-Hilbert space of the sort I've
secretly been talking about so far: that is, finite-dimensional ones.

So we may, perhaps, need to ponder representations of Lie 2-groups
on infinite-dimensional 2-Hilbert spaces.

Luckily, that's just what Derek Wise spoke about on Wednesday morning!
His talk also included some pictures with intriguing relations to
the pictures in Bruce's talk. You can see the slides here:

They make a nice introduction to a paper he's writing with Aristide
Baratin, Laurent Freidel and myself. Our work uses ideas like
measurable fields of Hilbert spaces, which are already important for
understanding infinite-dimensional unitary group representations. But
if you're less fond of analysis, jump straight to pages 20, 23 and 25,
where he gives a geometrical interpretation of these
infinite-dimensional representations, along with the intertwining
operators between them... and the "2-intertwining operators" between
those.

This work relies
heavily on the work of Crane, Sheppeard and Yetter, cited in "week210" - so check out that, too!

There's much more to say, but I'm running out of steam, so I'll
just mention a few more talks: Enrico Vitale's talk on categorified
homological algebra, and the talks by David Roberts and Aurora del
Río on the fundamental 2-group of a topological space.

To set these in their proper perspective, it's good to recall
the periodic table of n-categories, mentioned in "week49":

The idea here is that an (n+k)-category with only one j-morphism for j
< k acts like an n-category with extra bells and whistles: a "k-tuply
monoidal n-category". This idea has not been fully established, and
there are some problems with naive formulations of it, but it's bound
to be right when properly understood, and it's useful for anyone trying
to understand the big picture of mathematics.

Now, an n-category with everything invertible is called an
"n-groupoid". Such a thing is believed to be essentially
the same as a "homotopy n-type", meaning a nice space, like
a CW complex, with vanishing homotopy groups above the nth - where we
count homotopy equivalent spaces as the same. If we accept this, the
n-groupoid version of the Periodic Table can be understood using
homotopy theory. It looks like this:

Most of this workshop has focused on 2-groups. But abelian groups are
especially interesting and nice, and there's a huge branch of math
called "homological algebra" that studies categories similar to the
category of abelian groups. These are called "abelian categories".
In an abelian category, you've got direct sums, kernels, cokernels,
exact sequences, chain complexes and so on - all things you're used to
in the category of abelian groups!

Can we categorify all this stuff? Yes - and that's what Enrico Vitale
is busy doing! He started by telling us how all these ideas generalize
from abelian groups to symmetric 2-groups, and how they change.

For example, besides the "kernel" and "cokernel",
we also need extra concepts. The reason is that the kernel of a
homomorphism says if the homomorphism is one-to-one, while its
cokernel says if it's onto. Functions can be nice in two basic
ways: they can be one-to-one, or onto. But because categories have an
extra level, functors between them can be nice in three ways,
called "faithful", "full" and "essentially
surjective". So, we need more than just the kernel and cokernel
to say what's going on. We also need the "pip" and
"copip".

The concepts of exact sequence and chain complex get subtler, too.
You can read about these things here:

By generalizing properties of the category of abelian groups, people
invented the concept of "abelian category". Similarly,
Vitale told us a definition of "2-abelian 2-category",
obtained by generalizing properties of the 2-category of symmetric
2-groups. I believe this is discussed here:

Mathieu Dupont is defending his dissertation on June 30th. I hope he
puts it on the arXiv after that. (He did!)

All this stuff gets even more elaborate as we move to n-groups for
higher n. To some extent this is the subject of homotopy theory, but
one also wants a more explicitly algebraic approach. See for example:

23) Giuseppe Metere: The ziqqurath of exact sequences of n-groupoids,
Ph.D. Thesis, Università di Milano, 2008. Also available at
arXiv:0802.0800.

The relation between 2-groups and topology is made explicit using
the concept of "fundamental 2-group". Just as every space equipped
with a basepoint has a fundamental group, it has a fundamental 2-group.
And for a homotopy 2-type, this 2-group captures everything about the
space - at least if we count homotopy equivalent spaces as the same.

David Roberts prepared an excellent talk about the fundamental
2-group of a space for this workshop. Unfortunately, he was unable
to come. Luckily, you can still see his talk:

The basic principle of Galois theory says that covering spaces of
a connected space are classified by subgroups of its fundamental
group. Here Roberts explains how "2-covering spaces" of a connected
space are classified by "sub-2-groups" of its fundamental 2-group!

Aurora del Río spoke on fundamental 2-groups and their application
to K-theory. Whenever we have a fibration of pointed spaces

F → E → B

we get a long exact sequence of homotopy groups

... → πn(F) → πn(E) → πn(B) → πn-1(F) → ...

This is a standard tool in algebraic topology; I sketched how it
works in "week151".

Now, the nth homotopy group of a space X, written πn(X),
is just the fundamental group of the (n-1)-fold loop space of X. So,
the Spanish categorical group experts define the nth "homotopy
2-group" of a space X to be the fundamental 2-group of an
iterated loop space of X. And, it turns out that any fibration of
spaces gives a long exact sequence of homotopy 2-groups!

I was surprised by this, but in retrospect I shouldn't have been.
Any fibration gives a "long exact sequence of iterated loop spaces":

... → LnF → Ln E → Ln B →
Ln-1F → ...

So, as soon as we have a definition of "fundamental
n-groupoids" and long exact sequences of n-groupoids, and can
show that taking the fundamental n-groupoid preserves exactness,
we can get a long exact sequence of fundamental n-groupoids.
If we simply define a fundamental n-groupoid to be a homotopy
n-type, this should not be hard.

But this was just the warmup for Aurora's talk, which was about
K-theory. Quillen set up modern algebraic K-theory by defining the K-groups
of a ring R to be the homotopy groups of a certain space called
BGL(R)+. In here talk, Aurora defined the K-2-groups of a
ring in the same way, but using homotopy 2-groups! And then she went ahead and
studied them...

The slides for Aurora's talk are - as for many of the talks -
available from the workshop's website:

Actually Porter gave two talks. The first was an introduction to
simplicial methods and crossed complexes, but Bartlett didn't
summarize that, and no slides are available.
So for that, you should get ahold of the following
free book: